10 research outputs found
Close encounters of a rotating star with planets in parabolic orbits of varying inclination and the formation of Hot Jupiters
(abbreviated) We extend the theory of close encounters of a planet on a
parabolic orbit with a star to include the effects of tides induced on the
central rotating star. Orbits with arbitrary inclination to the stellar
rotation axis are considered. We obtain results both from an analytic treatment
and numerical one that are in satisfactory agreement. These results are applied
to the initial phase of the tidal circularisation problem. We find that both
tides induced in the star and planet can lead to a significant decrease of the
orbital semi-major axis for orbits having periastron distances smaller than 5-6
stellar radii (corresponding to periods days after the
circularisation has been completed) with tides in the star being much stronger
for retrograde orbits compared to prograde orbits. We use the simple Skumanich
law for the stellar rotation with its rotational period equal to one month at
the age of 5Gyr. The strength of tidal interactions is characterised by
circularisation time scale, defined as a time scale of evolution of
the planet's semi-major axis due to tides considered as a function of orbital
period after the process of tidal circularisation has been completed.
We find that the ratio of the initial circularisation time scales corresponding
to prograde and retrograde orbits is of order 1.5-2 for a planet of one Jupiter
mass and four days. It grows with the mass of the planet, being
of order five for a five Jupiter mass planet with the same . Thus, the
effect of stellar rotation may provide a bias in the formation of planetary
systems having planets on close orbits around their host stars, as a
consequence of planet-planet scattering, favouring systems with retrograde
orbits. The results may also be applied to the problem of tidal capture of
stars in young stellar clusters.Comment: to be published in Celestial Mechanics and Dynamical Astronom
Characterizing Multi-planet Systems with Classical Secular Theory
Classical secular theory can be a powerful tool to describe the qualitative
character of multi-planet systems and offer insight into their histories. The
eigenmodes of the secular behavior, rather than current orbital elements, can
help identify tidal effects, early planet-planet scattering, and dynamical
coupling among the planets, for systems in which mean-motion resonances do not
play a role. Although tidal damping can result in aligned major axes after all
but one eigenmode have damped away, such alignment may simply be fortuitous. An
example of this is 55 Cancri (orbital solution of Fischer et al., 2008) where
multiple eigenmodes remain undamped. Various solutions for 55 Cancri are
compared, showing differing dynamical groupings, with implications for the
coupling of eccentricities and for the partitioning of damping among the
planets. Solutions for orbits that include expectations of past tidal evolution
with observational data, must take into account which eigenmodes should be
damped, rather than expecting particular eccentricities to be near zero.
Classical secular theory is only accurate for low eccentricity values, but
comparison with other results suggests that it can yield useful qualitative
descriptions of behavior even for moderately large eccentricity values, and may
have advantages for revealing underlying physical processes and, as large
numbers of new systems are discovered, for triage to identify where more
comprehensive dynamical studies should have priority.Comment: Published in Celestial Mechanics and Dynamical Astronomy, 25 pages,
10 figure
The generalized non-conservative model of a 1-planet system - revisited
We study the long-term dynamics of a planetary system composed of a star and
a planet. Both bodies are considered as extended, non-spherical, rotating
objects. There are no assumptions made on the relative angles between the
orbital angular momentum and the spin vectors of the bodies. Thus, we analyze
full, spatial model of the planetary system. Both objects are assumed to be
deformed due to their own rotations, as well as due to the mutual tidal
interactions. The general relativity corrections are considered in terms of the
post-Newtonian approximation. Besides the conservative contributions to the
perturbing forces, there are also taken into account non-conservative effects,
i.e., the dissipation of the mechanical energy. This dissipation is a result of
the tidal perturbation on the velocity field in the internal zones with
non-zero turbulent viscosity (convective zones). Our main goal is to derive the
equations of the orbital motion as well as the equations governing
time-evolution of the spin vectors (angular velocities). We derive the
Lagrangian equations of the second kind for systems which do not conserve the
mechanical energy. Next, the equations of motion are averaged out over all fast
angles with respect to time-scales characteristic for conservative
perturbations. The final equations of motion are then used to study the
dynamics of the non-conservative model over time scales of the order of the age
of the star. We analyze the final state of the system as a function of the
initial conditions. Equilibria states of the averaged system are finally
discussed.Comment: 37 pages, 13 figures, accepted to Celestial Mechanics and Dynamical
Astronom
Tidal friction in close-in satellites and exoplanets. The Darwin theory re-visited
This report is a review of Darwin's classical theory of bodily tides in which
we present the analytical expressions for the orbital and rotational evolution
of the bodies and for the energy dissipation rates due to their tidal
interaction. General formulas are given which do not depend on any assumption
linking the tidal lags to the frequencies of the corresponding tidal waves
(except that equal frequency harmonics are assumed to span equal lags).
Emphasis is given to the cases of companions having reached one of the two
possible final states: (1) the super-synchronous stationary rotation resulting
from the vanishing of the average tidal torque; (2) the capture into a 1:1
spin-orbit resonance (true synchronization). In these cases, the energy
dissipation is controlled by the tidal harmonic with period equal to the
orbital period (instead of the semi-diurnal tide) and the singularity due to
the vanishing of the geometric phase lag does not exist. It is also shown that
the true synchronization with non-zero eccentricity is only possible if an
extra torque exists opposite to the tidal torque. The theory is developed
assuming that this additional torque is produced by an equatorial permanent
asymmetry in the companion. The results are model-dependent and the theory is
developed only to the second degree in eccentricity and inclination
(obliquity). It can easily be extended to higher orders, but formal accuracy
will not be a real improvement as long as the physics of the processes leading
to tidal lags is not better known.Comment: 30 pages, 7 figures, corrected typo
The Relativistic Factor in the Orbital Dynamics of Point Masses
There is a growing population of relativistically relevant minor bodies in
the Solar System and a growing population of massive extrasolar planets with
orbits very close to the central star where relativistic effects should have
some signature. Our purpose is to review how general relativity affects the
orbital dynamics of the planetary systems and to define a suitable relativistic
correction for Solar System orbital studies when only point masses are
considered. Using relativistic formulae for the N body problem suited for a
planetary system given in the literature we present a series of numerical
orbital integrations designed to test the relevance of the effects due to the
general theory of relativity in the case of our Solar System. Comparison
between different algorithms for accounting for the relativistic corrections
are performed. Relativistic effects generated by the Sun or by the central star
are the most relevant ones and produce evident modifications in the secular
dynamics of the inner Solar System. The Kozai mechanism, for example, is
modified due to the relativistic effects on the argument of the perihelion.
Relativistic effects generated by planets instead are of very low relevance but
detectable in numerical simulations
Dense Stellar Populations: Initial Conditions
This chapter is based on four lectures given at the Cambridge N-body school
"Cambody". The material covered includes the IMF, the 6D structure of dense
clusters, residual gas expulsion and the initial binary population. It is aimed
at those needing to initialise stellar populations for a variety of purposes
(N-body experiments, stellar population synthesis).Comment: 85 pages. To appear in The Cambridge N-body Lectures, Sverre Aarseth,
Christopher Tout, Rosemary Mardling (eds), Lecture Notes in Physics Series,
Springer Verla
Tides in Planetary Systems
International audienceThe Solar system is the seat of many interactions between the Sun, the planets and their natural satellites. Moreover, since 1995, a large number of extrasolar planetary systems has been discovered where planets orbit around other stars, sometimes very close to them. Therefore, in such systems, tidal interactions are one of the key mechanisms that must be studied to understand the celestial bodies' dynamics and evolution. Indeed, tides generate displacements and flows in planetary (and in the host star) interiors. The associated kinetic energy is then dissipated into heat because of internal friction processes. This leads to secular evolution of orbits and of spins with characteristic time-scales that are intrinsically related to the properties of dissipative mechanisms, those latters depending both on the internal structure of the studied bodies and on the tidal frequency. This lecture is aimed to review the must advanced theories to study tidal dynamics in planetary systems and the different tidal flows or displacements that can be excited by a perturber, the conversion of their kinetic energy into heat, the related exchanges of angular momentum, and the consequences for systems evolution